You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
Mathematical Expression Editor
[?]
Illegal control sequence name for \newcommand
Objectives:
1.
Understand what a vector valued function is.
2.
Be able to do calculus with vector valued functions: differentiation,
integration.
3.
Understand what the derivative of a vector valued function
represents geometrically.
4.
Be able to find the arc length of a curve between two points.
Recap Video
You can watch watch the following video which recaps the ideas of the section.
Test your understanding with the following questions.
Parametrization Examples
Below are three videos showing examples of parametrizing curves.
Parametrize the graph in the -plane of .
Parametrize the intersection of the cylinder and the plane .
Parametrize the ellipse .
Problems
Parametrize the intersection of the parabolic sheet with the plane .
If we let ,
then the parametrization becomes
Parametrize the curve in 3D given by in the plane .
We know we can
parametrize as and . Therefore, the parametrization for the curve
is
Suppose is the curve given by .
The vector .
The derivative .
A tangent vector to at the point is . Plugging this into the
derivative gives the vector
Suppose . If , then
Suppose a particle is moving with position function . At what value of is the
speed of the particle smallest?
Steps:
The velocity of the particle is given by .
The speed of the particle is
The speed is smallest at the -value:
Find the arc length of the curve between the points and .
Steps:
The -value corresponding to the point is
The -value corresponding to the point is
The velocity vector is .
The speed is
The arc length is
The arc length of the curve between the points and is: .
The expression
factors as .
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
Controls
Press...
...to do
left/right arrows
Move cursor
shift+left/right arrows
Select region
ctrl+a
Select all
ctrl+x/c/v
Cut/copy/paste
ctrl+z/y
Undo/redo
ctrl+left/right
Add entry to list or column to matrix
shift+ctrl+left/right
Add copy of current entry/column to to list/matrix
ctrl+up/down
Add row to matrix
shift+ctrl+up/down
Add copy of current row to matrix
ctrl+backspace
Delete current entry in list or column in matrix
ctrl+shift+backspace
Delete current row in matrix
×
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)